since it is this combination that appears in the canonically normalised propagator.
In four dimensions the integral \( \int {\rm {d}}^4x \sqrt{\det g} E\)  is a topological invariant, so \(c_k\)  will not enter into any derivative of \( \bar{\Gamma }_k[g] \)  and, as such, \(c_k\)  does not appear in any beta function [1]}, [2]}.
